HYACINTHOS 28391

 

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and L1, L2, L3 three lines perpendiculars to BC, CA, AB, resp

Denote:

Na, Nb, Nc = NPC centers of NBC,NCA, NAB, resp.

R1, R2, R3 =  the reflections of L1, L2, L3 in NbNc, NcNa, NaNb, resp.

A*B*C* = the triangle bounded by R1, R2, R3

ABC, A*B*C* are parallelogic and the parallelogic center (ABC, A*B*C*) is independent from the lines L1, L2. L3

It is X(24144)  

Application:

Let ABC be a triangle P a point and A'B'C' the pedal triangle of P.

Denote:

Na, Nb, Nc = NPC centers of NBC,NCA, NAB, resp.

R1, R2, R3 =  the reflections of PA', PB', PC' in NbNc, NcNa, NaNb, resp.

A*B*C* = the triangle bounded by R1, R2, R3

ABC, A*B*C* are parallelogic.

Parallelogic center (ABC, A*B*C*) = X(24144)  

 Special cases:

1. P = O

2. P = H

Parallelogic centers (A*B*C*, ABC) ?

[Peter Moses]:


Hi Antreas,

1). 6 a^16-33 a^14 b^2+75 a^12 b^4-89 a^10 b^6+55 a^8 b^8-11 a^6 b^10-7 a^4 b^12+5 a^2 b^14-b^16-33 a^14 c^2+102 a^12 b^2 c^2-97 a^10 b^4 c^2+4 a^8 b^6 c^2+41 a^6 b^8 c^2-14 a^4 b^10 c^2-7 a^2 b^12 c^2+4 b^14 c^2+75 a^12 c^4-97 a^10 b^2 c^4+8 a^8 b^4 c^4-3 a^6 b^6 c^4+30 a^4 b^8 c^4-9 a^2 b^10 c^4-4 b^12 c^4-89 a^10 c^6+4 a^8 b^2 c^6-3 a^6 b^4 c^6-18 a^4 b^6 c^6+11 a^2 b^8 c^6-4 b^10 c^6+55 a^8 c^8+41 a^6 b^2 c^8+30 a^4 b^4 c^8+11 a^2 b^6 c^8+10 b^8 c^8-11 a^6 c^10-14 a^4 b^2 c^10-9 a^2 b^4 c^10-4 b^6 c^10-7 a^4 c^12-7 a^2 b^2 c^12-4 b^4 c^12+5 a^2 c^14+4 b^2 c^14-c^16 : : 

 

= 3 X[1157] - X[14072], X[137] - 3 X[24147].  

= lies on these lines: {5,16764},{30,11671},{137,24147},{195,10126},{1157,14072},{5965,6592},{14143,22051}.


2). 2 a^16-11 a^14 b^2+25 a^12 b^4-27 a^10 b^6+5 a^8 b^8+23 a^6 b^10-29 a^4 b^12+15 a^2 b^14-3 b^16-11 a^14 c^2+34 a^12 b^2 c^2-35 a^10 b^4 c^2+12 a^8 b^6 c^2-21 a^6 b^8 c^2+62 a^4 b^10 c^2-61 a^2 b^12 c^2+20 b^14 c^2+25 a^12 c^4-35 a^10 b^2 c^4+8 a^8 b^4 c^4+7 a^6 b^6 c^4-38 a^4 b^8 c^4+93 a^2 b^10 c^4-60 b^12 c^4-27 a^10 c^6+12 a^8 b^2 c^6+7 a^6 b^4 c^6+10 a^4 b^6 c^6-47 a^2 b^8 c^6+108 b^10 c^6+5 a^8 c^8-21 a^6 b^2 c^8-38 a^4 b^4 c^8-47 a^2 b^6 c^8-130 b^8 c^8+23 a^6 c^10+62 a^4 b^2 c^10+93 a^2 b^4 c^10+108 b^6 c^10-29 a^4 c^12-61 a^2 b^2 c^12-60 b^4 c^12+15 a^2 c^14+20 b^2 c^14-3 c^16 : : 

 

= lies on these lines: {2,3},{137,24147},{1263,19552},{3448,11584},{5663,24043},{8254,20414}}.
= midpoint of X(1263) and X(19552).
= {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 5, 20030), (5, 10205, 140), (3850, 10289, 5).

Best regards,
Peter Moses.